6th Maths NCERT Chapter 6
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Important Topics to Study:
1. Rate of Change of Quantities
– Summary: The derivative represents the rate of change of a quantity with respect to another. It is used to determine rates of change, equations of tangents and normals to curves, turning points, intervals of increase or decrease, and approximate values of quantities.
2. Applications in Real-World Problems
– Summary: Derivatives have practical applications in areas like calculating rates of change for geometric shapes, finding areas of circles, volumes of objects, and speeds of waves in circles.
3. Marginal Cost and Revenue
– Summary: Understanding marginal cost and revenue involves finding the instantaneous rate of change of total cost and total revenue with respect to the quantity produced or sold, respectively.
Additional Concepts for Study:
– Using derivatives to find rates of change in various scenarios such as circle areas, cube volumes, and wave areas.
– Applying the Chain Rule to find relationships between different variables and their rates of change.
– Calculating rates of increase in perimeters and areas of shapes with changing dimensions.
These topics cover the practical application of derivatives in various contexts and provide a foundation for solving real-world problems using calculus.
Explain what the derivative represents when one quantity y varies with another quantity x.
Calculate the rate of change of the area of a circle per second with respect to its radius r when r = 5 cm.
When a stone is dropped into a quiet lake, and waves move in circles at a speed of 4 cm per second, how fast is the enclosed area increasing when the radius of the circular wave is 10 cm?
A rectangle has a length decreasing at 3 cm/min and a width increasing at 2 cm/min. Find the rates of change of (a) the perimeter and (b) the area of the rectangle when x = 10 cm and y = 6 cm.
Calculate the marginal cost when 3 units are produced, given the total cost function C(x) = 0.005x^3 – 0.02x^2 + 30x + 5000.
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